Perimeter and Circumference: Measuring Boundaries

Equilateral Triangles

An equilateral triangle is a special type of triangle where all three sides are exactly the same length. If we let $s$ represent the length of one side, the formula for the perimeter is straightforward: $P = 3s$

Worked Example: The Garden Fence

Suppose you have a rectangular garden that is 10 meters long and 5 meters wide. To find how much fencing you need, apply the rectangle formula: $P = 2(10\text{ m} + 5\text{ m})$ $P = 2(15\text{ m})$ $P = 30\text{ m}$

Measuring the Curve: Circumference

When we move from shapes with straight edges to curved shapes like circles, the term "perimeter" is replaced by circumference. Because a circle has no corners or straight sides to add up, we rely on a special mathematical constant called $\pi$ (pi).

$\pi$ represents the ratio of a circle's circumference to its diameter. It is an irrational number, approximately equal to $3.14159...$, though in many classroom settings, using $3.14$ or the fraction $\frac{22}{7}$ is sufficient.

The Formulas for Circumference

To calculate the circumference ($C$), you need to know either the radius ($r$) or the diameter ($d$) of the circle.

1. Using Diameter: The diameter is the distance across the circle passing through the center. The formula is: $C = \pi \cdot d$
2. Using Radius: The radius is the distance from the center to the edge. Since the diameter is exactly twice the radius ($d = 2r$), the formula becomes: $C = 2 \cdot \pi \cdot r$

Worked Example: The Circular Track

A circular running track has a radius of 50 meters. What is the distance of one lap around the track? Using the radius formula: $C = 2 \cdot \pi \cdot 50$ $C = 100\pi$ If we use the approximation $\pi \approx 3.14$: $C \approx 100 \cdot 3.14$ $C \approx 314\text{ meters}$

Pro Tip: If a problem provides the diameter but asks for the answer in terms of $\pi$, do not multiply by 3.14. Simply multiply the diameter by $\pi$ and leave the symbol in your answer (e.g., $10\pi$).

Summary Table of Formulas

To help you study, here is a quick reference guide for the shapes discussed in this article:

By mastering these formulas, you can accurately measure the boundaries of almost any standard geometric shape you encounter in geometry or real-world applications.